Question

Select the solutions for the following:

2(x+4)=3(x+12)

7(2+5v=3v+14

a
one solution

b
no solutions

c
infinate solutions

Answer 1

Let's solve each equation step by step to determine the number of solutions.

### 1. Solve the equation \(2(x + 4) = 3(x + 12)\):

First, distribute on both sides:

\[
2x + 8 = 3x + 36
\]

Next, rearranging the equation to isolate terms involving \(x\):

\[
2x + 8 - 3x = 36
\]

Combine like terms:

\[
-x + 8 = 36
\]

Subtract 8 from both sides:

\[
-x = 36 - 8
\]

\[
-x = 28
\]

Multiply by -1:

\[
x = -28
\]

Since we found a specific value for \(x\), this equation has **one solution**.

### 2. Solve the equation \(7(2 + 5v) = 3v + 14\):

First, distribute the 7 on the left side:

\[
14 + 35v = 3v + 14
\]

Now, let's rearrange to isolate \(v\):

Subtract \(3v\) from both sides:

\[
14 + 35v - 3v = 14
\]

Combine like terms:

\[
14 + 32v = 14
\]

Now, subtract 14 from both sides:

\[
32v = 0
\]

Divide by 32:

\[
v = 0
\]

Since we found a specific value for \(v\), this equation also has **one solution**.

### Summary of Solutions:
1. **For \(2(x + 4) = 3(x + 12)\)**: One solution
2. **For \(7(2 + 5v) = 3v + 14\)**: One solution

**Final Answers**:
- Both equations have **one solution** (Option a).